Margin of Error Calculator

MOE = z × √(p(1−p) ÷ n), expressed in percentage points.

z comes from your confidence level: 1.96 for 95%, 1.645 for 90%, 2.576 for 99%.

p = 50% is the default because it produces the widest margin. If your actual result is far from 50%, the real margin is narrower than the headline figure.

When you supply a population size, the finite population correction √((N − n) ÷ (N − 1)) is applied — it matters once your sample is more than about 5% of the population.

For a survey percentage, the margin of error is MOE = z × √(p(1−p) ÷ n). n is the number of completed responses, p is the result as a decimal, and z comes from the confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%. The calculator above also offers 80% and 85%. If you enter a population size, it multiplies the result by the finite population correction √((N − n) ÷ (N − 1)); leave the field blank and it assumes a very large population.

The margin is expressed in percentage points. A result of 58% with a margin of ±3 means 55% to 61%, not 58% plus or minus 3% of 58.

At 95% confidence with the cautious p = 50%: 1.96 × √(0.25 ÷ 400) = 1.96 × 0.025 = 0.049, so ±4.9 points. That is the calculator's default setting.

If the result was actually 20%: 1.96 × √(0.16 ÷ 400) = 1.96 × 0.02 = ±3.9 points. At 10% it narrows to ±2.9. The headline figure pollsters quote uses 50% because it is the widest case, so it covers every answer in the survey.

If those 400 responses came from a population of only 2,000: 4.9 × √(1,600 ÷ 1,999) = 4.9 × 0.895 = ±4.4 points. The correction starts to matter once your sample is more than about 5% of the population.

They are three views of the same quantity. The standard error is √(p(1−p) ÷ n): how much the estimate would bounce around from sample to sample. The margin of error is the standard error times z, so it depends on the confidence level you chose. The confidence interval is the estimate plus and minus the margin: 60% ± 4.9 is an interval of 55.1% to 64.9%.

Standard deviation is different again. It describes how spread out individual answers are (for a rating question, say), while the standard error describes how precise the average of those answers is. Margin of error calculators that ask for a standard deviation are working with means, not percentages; this one works with percentages.

One caveat about the range shown above the guide: it is the simple symmetric one, the result plus and minus the margin, cut off at 0% and 100%. For small samples or results near 0% or 100%, that range is unreliable; the confidence interval calculator uses the Wilson score method, which handles those cases.

Bigger than either group's own margin — a point most calculators skip. For two independent groups (customers vs non-customers, this year's survey vs last year's), the margin on the difference is roughly √(MOE₁² + MOE₂²).

Illustration: 62% of 300 customers and 48% of 250 non-customers agree with a statement. Their margins at 95% are ±5.5 and ±6.2 points. The margin on the 14-point gap is √(5.5² + 6.2²) = ±8.3 points, so the difference is larger than the noise. A 6-point gap with the same samples would not be. For two equal-sized groups, the difference margin is about 1.4 times the single-group margin.

Within a single question, comparing two answers that compete with each other (candidate A vs candidate B, option 1 vs option 2) is worse still. Pew Research Center notes that the margin on the lead is generally about twice the margin for each candidate: a poll of about 1,000 with ±3 points per candidate has roughly ±6 on the gap. A 4-point lead in that poll is not a clear lead. For a formal test between two groups, the A/B test significance calculator gives a p-value.

Because the margin depends on the number of responses behind each figure, and a subgroup has fewer. AAPOR's example: a survey of 1,000 adults with ±3 points overall that includes 200 Hispanic respondents has a margin of about ±6.9 points for results about that group. Run the calculator with 200 and you get the same ±6.9.

Pew Research Center shows margins of error in charts when a subgroup's effective sample size is under 100, and does not report subgroups with fewer than 100 interviews at all. Applying the same rule to a staff survey means not publishing a team-level result built on 14 replies — it is too noisy to act on, and in small teams it can also identify people.

It measures random sampling error only. Question wording, answer order, who chose not to respond and people misremembering are not in it. AAPOR states plainly that there is no measurable overall margin of error for a poll, because those other errors cannot be measured.

Weighting also widens it. If you weight responses to match the population, the effective sample size shrinks by the design effect. Illustration: 1,000 responses with a design effect of 1.5 behave like 667, which puts the margin at about ±3.8 rather than ±3.1. High-quality surveys fold this into the figure they publish; the calculator here assumes a simple random sample.

And it only applies to probability samples. AAPOR says the margin of sampling error does not apply to opt-in online surveys and other non-probability polls. Pew Research Center's 2023 comparison found opt-in samples averaged 5.8 points of error against benchmarks, versus 2.6 for probability-based panels. If your survey went out as a public link, report the number of responses and how people found the survey, not a margin of error.

The margin is only as good as the sample. Send the survey to a defined list rather than posting it publicly, turn on one response per person so nobody is counted twice, and use hidden fields to tag which invitation each response came from. Zunoform can also randomize answer-option order per respondent, which reduces the pull of whichever option is listed first, and close the form automatically at a response count so fieldwork ends at the planned sample. The Free plan includes 500 responses a month.